5: What Is a Vector?

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"A quantity that has both size and direction"? But what is 'size' or 'direction'? In fact, any vector is an element of a vectors space, which is not any tautology.

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Topics

About: high school mathematics

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: A Vector Is "a Quantity That Has Both Size and Direction"?
• 2: Enter 'Scalar Multiplication'!
• 3: Vector Is an Element of a Vectors Space
• 4: What 'Size' Really Means
• 5: How Is the Set of the n-Tuples of Real Numbers Related to a Vectors Space?
• 6: What Is the Dimension of a Vectors Space?

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Starting Context

• The reader knows the background of this site.

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Target Context

• The reader will know what a vector is.

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Orientation

There is an article on becoming a benefactor of humanity by being a conduit of truths

There is an article on definition of set and Russel's "paradox".

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Main Body

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1: A Vector Is "a Quantity That Has Both Size and Direction"?

Special-Student-7-Hypothesizer
The most prevalent rough definition of 'vector' seems to be "any quantity that has both size and direction".

Special-Student-7-Rebutter
That is not any valid definition unless "size" and "direction" are defined.

Special-Student-7-Hypothesizer
'size' is ... a size, you know.

Special-Student-7-Rebutter
I do not know, actually.

What is 'size' of a book? The length?, the width?, the thickness?, the weight?, the number of pages?, the number of words?, the amount of the information contained in it?, the price?, or what? Or why cannot I just declare it to be 42?

Special-Student-7-Hypothesizer
I do not know why you cannot: if you define the size to be 42, the size should be 42.

Special-Student-7-Rebutter
Then, I do not understand what "has size" means, because anything can be regarded to "have size" by just our regarding it to have "size 42".

Special-Student-7-Hypothesizer
That seems true.

Special-Student-7-Rebutter
And what is 'direction'?

Special-Student-7-Hypothesizer
In the 3-dimensional space, you know what 'direction' is.

Special-Student-7-Rebutter
For the 3-dimensional space case, yes, I have some intuition, but that is not any general definition.

Special-Student-7-Hypothesizer
Well, when the quantity is a combination of 4 real numbers like (1.3, 2.5, 0.9, 4.2), the direction is the proportion of the 4 numbers: (1.3, 2.5, 0.9, 4.2) and (1.3 * 2, 2.5 * 2, 0.9 * 2, 4.2 * 2) have the same direction.

Special-Student-7-Rebutter
Is any vector always a combination of some real numbers?

Special-Student-7-Hypothesizer
That is not the case.

Special-Student-7-Rebutter
Then, that is just an example, not any general definition.

Special-Student-7-Hypothesizer
That is true.

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2: Enter 'Scalar Multiplication'!

Special-Student-7-Hypothesizer
In fact, we need to introduce the concept of 'scalar multiplication' in order to begin to talk about 'direction'.

We introduce a scalars set, which is a field, and introduce a scalar multiplication, which means multiplying a vector by an element of the scalars set to get a vector.

Any field is, very roughly speaking, a structure that allows a set of arithmetic operations, (+, -, *, /), and a typical example is the real numbers set, $\mathbb{R}$.

Just for the sake of simplicity, in this article, we will assume that the scalars set is always the real numbers set unless specified otherwise.

Any vector, v, can be multiplied by any real number, r, and the result is r v.

In the above example, v = (1.3, 2.5, 0.9, 4.2), r = 2, and r v = (1.3 * 2, 2.5 * 2, 0.9 * 2, 4.2 * 2).

And for any fixed v, r v with any r has the same direction.

On the other hand, for some 2 vectors, v and v', if there is no r such that v' = r v, v and v' have some different directions.

You can say that any direction is an equivalence class of vectors with respect to the equivalence relation, v ~ v' if and only if v = r v'.

Special-Student-7-Rebutter
When r < 0, do v and r v have the same direction?

Special-Student-7-Hypothesizer
Well, that is a matter of the definition of the word, and we can say that they have the same direction, while colloquially, they are usually said to have "opposite directions" (the problem of calling it "opposite directions" is that the scalars set is not necessarily the real numbers set and the scalars set does not have necessarily have the concept of "plus" or "minus").

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3: Vector Is an Element of a Vectors Space

Special-Student-7-Rebutter
Anyway, a moral is that it is meaningless to talk about whether a single object has a direction by itself: we need to think of a space of objects on which a scalar multiplication is defined, and only by dividing the space into equivalence classes, the concept of direction arises.

Without 1st defining the space, we cannot talk about whether an object "has a direction", so, it is meaningless to talk about whether a single object is a vector by itself.

Special-Student-7-Hypothesizer
So, we say that any vector is an element of a vectors space.

Someone may say that that is a tautology, but no, we 1st define a space and call the space a vectors space, and then, an element of the space is called "vector".

A rigorous definition of vectors space is here.

Special-Student-7-Rebutter
The way of thinking is important, because it is prevalent in mathematics.

For example, mathematics talks about spaces (or called "structures") like groups, rings, fields, modules, etc., and a group element is an element of a group: it is meaningless to talk about a group element by itself.

The prevalent pattern is, we take a set and take some operations on the set with some rules, and the set with the operations is a space (structure).

'vectors space' is one of such spaces and a vector is an element of such a space.

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4: What 'Size' Really Means

Special-Student-7-Rebutter
That rigorous definition does not mention 'size' at all.

Special-Student-7-Hypothesizer
In fact, a vector does not need to have any absolute size.

Special-Student-7-Rebutter
What do you mean by "absolute size"?

Special-Student-7-Hypothesizer
For a nonzero vector, v, and a positive scalar, r, the "size" of r v could be said to be r-times the "size" of v.

So, we could talk about the relative sizes of some 2 vectors in the same direction, but comparing the sizes of some 2 vectors in some different directions is not meaningful in general.

For example, think of a vectors space, in which any element is the tuple of the length (in cm) and the weight (in kg) of an object.

Special-Student-7-Rebutter
Strictly speaking, that is not any vectors space, because each negative scalar multiple of any vector must exist in the space.

Special-Student-7-Hypothesizer
Then, think of differences of 2 lengths and differences of 2 weights, which can be negative.

Special-Student-7-Rebutter
All right.

Special-Student-7-Hypothesizer
Is the size of a vector, (180, 70), meaningful?

Special-Student-7-Rebutter
I, personally, find it not so meaningful.

Special-Student-7-Hypothesizer
And which is larger between (185, 65) and (175, 75)?

Special-Student-7-Rebutter
Someone may say that (175, 75) < (185, 65) claiming that length is more important, another one may say that (185, 65) < (175, 75) claiming that weight is more important, and yet another one may compare between ${185}^2+{65}^2$ and ${175}^2+{75}^2$, while I, personally, do not believe that ${185}^2+{65}^2$ is particularly meaningful (for example, if we take length in mm instead, the comparison can be changed).

Special-Student-7-Hypothesizer
Or while the set of at-most-n-degree real polynomials canonically constitutes a vectors space, what is the "size" of a polynomial?

Special-Student-7-Rebutter
You could define a "size", for example, as the square root of the sum of the squared coefficients, or as the absolute value of the n-degree coefficient, or as uniformly 42, why not?

Special-Student-7-Hypothesizer
A vector does not have any intrinsic "size" in general, even if we could compare the sizes of some 2 vectors in the same direction.

Special-Student-7-Rebutter
What if the scalars set is not the real numbers set?

Special-Student-7-Hypothesizer
Then, even talking about the relative sizes of some 2 vectors in the same direction may be meaningless.

Special-Student-7-Rebutter
Then, what does "any quantity that has both size and direction" mean?

Special-Student-7-Hypothesizer
"has size" seems to mean that a vector can be multiplied by scalars to become different vectors in the same direction.

Special-Student-7-Rebutter
There is the concept of 'norm' and you need to be careful not to confuse 'norm' with "size" meant in the meaning mentioned above.

Special-Student-7-Hypothesizer
In fact, "size" usually meant is really 'norm', but a vectors space does not need to be equipped with a norm.

I guess that "size" in "any quantity that has both size and direction" is prevalently understood as 'norm', but in that meaning, "vector is any quantity that has both size and direction" is wrong.

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5: How Is the Set of the n-Tuples of Real Numbers Related to a Vectors Space?

Special-Student-7-Hypothesizer
Someone may think that any n-dimensional real vectors space is nothing but the set of the n-tuples of real numbers.

That is not exactly true, although any n-dimensional real vectors space can be represented by the set of the n-tuples of real numbers.

Special-Student-7-Rebutter
What does "be represented" mean?

Special-Student-7-Hypothesizer
Any n-dimensional real vectors space has a basis with some n elements, $b_1,...,b_n$, each vector, v, can be expressed as a linear combination of the basis elements, $v=v^1{b}_1+...+v^n{b}_n$, and the vector can be represented by the n-tuple of the coefficients, $(v^1,...,v^n)$.

"be represented" means that there is the 1-to-1 (bijective) linear mapping from the vectors space onto the set of the n-tuples of real numbers (technically, it is called that the vectors space is 'vectors spaces - linear morphisms' isomorphic to the set of the n-tuples of real numbers regarded as a vectors space).

Special-Student-7-Rebutter
Why does someone say as though any n-dimensional real vectors space is nothing but the set of the n-tuples of real numbers?

Special-Student-7-Hypothesizer
2 vectors spaces' being 'vectors spaces - linear morphisms' isomorphic means that the 2 vectors spaces have the same structure as vectors spaces, and the 2 vectors spaces can be handled in the same way as far as the vectors space operations are concerned.

Special-Student-7-Rebutter
"can be handled in the same way" does not mean that the 2 entities are the same entity.

Special-Student-7-Hypothesizer
Exactly, but some people like wordings like "the 2 spaces are the same", because probably those wordings bring about some labor-saving of expressions because expressions do not need to distinguish the 2 things, but at the core, we need to be aware that the 2 spaces are different.

For example, when we think of the change of the components of a vector with respect to a change of bases, unless you understand the existence of the vectors space independent of any representation, the concept would not make sense: as there is the vectors space independent of any representation, you return from a representation to the vectors space and go from the vectors space to another representation.

Not being able to distinguish between 'represented' and 'representation' seems a serious problem somehow prevalently seen.

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6: What Is the Dimension of a Vectors Space?

Special-Student-7-Rebutter
While we have talked about "n-dimensional real vectors space", what is 'dimension' of a vectors space?

Special-Student-7-Hypothesizer
'dimension' of any vectors space is nothing but the number of the elements of any basis for the vectors space.

Special-Student-7-Rebutter
In the 1st place, does every vectors space have a basis?

Special-Student-7-Hypothesizer
In fact, if you do not conceive that question, you are not really learning mathematics.

And the answer is yes, although we do not show any proof here.

Special-Student-7-Rebutter
And do all the possible bases of a vectors space have the same number of elements?

Special-Student-7-Hypothesizer
Also that question is one that if you do not conceive, you are not really learning mathematics.

And the answer is yes, although we do not show any proof here.

Let us know that a vectors space may be infinite-dimensional.

Special-Student-7-Rebutter
What does that mean?

Special-Student-7-Hypothesizer
That of course means that the number of the elements of any basis for the vectors space is infinite, but note that that does not mean that a vector is expressed as an infinite linear combination of basis elements: any vector is expressed as a finite linear combination of basis elements.

Special-Student-7-Rebutter
Then, why does the basis need some infinite number of elements?

Special-Student-7-Hypothesizer
A vector is a linear combination of a set of some finite number of basis elements, another vector is a linear combination of another set of some finite number of basis elements, and in order for every vector to be a linear combination of its own set of some finite number of basis elements, the basis needs some infinite number of elements.

Special-Student-7-Rebutter
While usually the space around us is said to be 3-dimensional, does that mean that the space is a 3-dimensional vectors space?

Special-Student-7-Hypothesizer
Aside from the fact that the spacetime is said to be 4-dimensional, strictly speaking, no, the space is not any 3-dimensional vectors space, which is because the space is said to be bounded.

Special-Student-7-Rebutter
Ah, so, the dimension aside, the space is not even any vectors space.

Special-Student-7-Hypothesizer
The space (or the spacetime) is modeled as a manifold, and the dimension of manifold has another definition, although it is somehow related with dimension of vectors space.

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References

<The previous article in this series | The table of contents of this series |

4: Are You Talking About Sets or About Collections?

<The previous article in this series | The table of contents of this series | The next article in this series>

Japanese schools talk about "shugo"s, but are they sets or collections?

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Topics

About: elementary school mathematics

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: What Does "shugo" (Used in Japanese Schools) Mean?
• 2: Caution: There Are Some Multiple Set Theories
• 3: How Have 'Set' and 'Collection' Ended Up Meaning Different Things?
• 4: In Fact, a Collection in General Is Not Defined by Any Property of Member
• 5: "shugo" Means 'Collection'
• 6: The Concept of 'Collection' Lives on
• 7: Mathematics Is Not Exactly Built on the Set Theory

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Starting Context

• The reader knows the background of this site.

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Target Context

• The reader will know the distinction between sets and collections.

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Orientation

There is an article on becoming a benefactor of humanity by being a conduit of truths

There is an article on definition of set and Russel's "paradox".

---

Main Body

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1: What Does "shugo" (Used in Japanese Schools) Mean?

Special-Student-7-Hypothesizer
Japanese schools talk about "shugo" s.

For example, they think of the "shugo" of some 3 apples, the "shugo" of the students of a class, etc..

Special-Student-7-Rebutter
What is "shugo" exactly?

Special-Student-7-Hypothesizer
Any "shugo" is any collection of some objects such that the membership is unambiguous.

Special-Student-7-Rebutter
Is there ever a collection such that the membership is ambiguous?

Special-Student-7-Hypothesizer
I do not think there is: the concept of 'collection' itself includes the requirement that the membership is unambiguous; the qualification is just for emphasis.

Special-Student-7-Rebutter
So, is "shugo" nothing but 'collection' in English?

Special-Student-7-Hypothesizer
I can just guess so, because Japanese textbooks do not say what "shugo" is called in English, but dictionaries usually say that "shugo" in mathematics is "set" in English.

Special-Student-7-Rebutter
But 'set' and 'collection' are 2 different things in mathematics.

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2: Caution: There Are Some Multiple Set Theories

Special-Student-7-Hypothesizer
As a caution, there are some multiple set theories, and what 'set' is in each theory depends on the theory.

Special-Student-7-Rebutter
Very confusing; cannot the theories use different names?

Special-Student-7-Hypothesizer
It is very confusing, but as each theory exists because it regards the other theories unsatisfactory, each theory says "'set' we call is the real 'set'!", and each theory uses "set".

Special-Student-7-Rebutter
I kind of understand, but ...

Special-Student-7-Hypothesizer
Anyway, arguably the most popular one is the ZFC set theory, and we will mean the ZFC set theory by "the set theory" and mean 'set' in the ZFC set theory by "set" hereafter.

Note that what we are going to say here are not particularly canonical; as we have not seen any convincing argument, we are presenting a hypothesis that is convincing for us. Nobody should accept something just because someone (whoever he or she is) says so or just because many people say so; check yourself whether the hypothesis is unboundedly consistent, which is the only way to approach truths.

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3: How Have 'Set' and 'Collection' Ended Up Meaning Different Things?

Special-Student-7-Hypothesizer
Most textbooks cite Russel's "paradox" as the reason why the set theory needs to be as it is, but the arguments are not convincing at all, at least for us.

Special-Student-7-Rebutter
What are the arguments like?

Special-Student-7-Hypothesizer
Historically, there was the so-called "naive set theory", which is based on the so-called "naive comprehension axiom", which is "Any precisely specified property of member can be used to define a set.".

Then, Russel's "paradox" came along and refuted the "naive comprehension axiom", and so refuted the "naive set theory".

So, now, the ZFC set theory admits only the empty set and the things constructed from the empty set as "set".

So, the collection of some 2 electrons are not admitted to be "set", because the collection is not constructed from the empty set.

Special-Student-7-Rebutter
There seems a wide gap in the reasoning: why would the collection of the 2 electrons not be "set" just because the "naive comprehension axiom" has been refuted?

Special-Student-7-Hypothesizer
I have not seen any convincing explanation.

In fact, why will we not have the modified axiom that "Any precisely specified property of member that (the property) determines membership unambiguously can be used to define a set."?

Special-Student-7-Rebutter
What Russel's "paradox" says is that just "precisely specified property of member" does not guarantee the unambiguous-ness of membership, so, why will we not add the unambiguous-ness as an additional requirement?

Special-Student-7-Hypothesizer
I have not seen any convincing argument why; it seems just that as checking it is not easy in general, they want a way to be able to define sets without bothering to do checking.

Special-Student-7-Rebutter
Well, is that an attitude to humbly approach truths? That seems an attitude to ignore whatever are inconvenient for humans.

Special-Student-7-Hypothesizer
Whether the unambiguous-ness can be checked easily is just a matter of human convenience, and the essence of the ZFC set theory is that it gave up covering the concept of "collection" and has decided to degenerate to deal with only convenient-for-humans kind of collections, which are now called "sets", which is my understanding.

In fact, I do not say at all that having a theory about such limited kind of collections is bad, but we need to be aware what the theory is really doing.

Special-Student-7-Rebutter
We need to avoid some confusions: "naive set theory" means the theory with the "naive comprehension axiom" not the concept of 'collection'; while the ZFC set theory excludes most collections like a collection of some 2 electrons, that is not because the collection of 2 electrons are inappropriate but because the set theory has degenerated.

Special-Student-7-Hypothesizer
The main cause of confusions is that the ZFC set theory keeps using the term, "set", while the concept of 'set' has degenerated from equaling 'collection' to 'convenient-for-humans collection'.

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4: In Fact, a Collection in General Is Not Defined by Any Property of Member

Special-Student-7-Hypothesizer
In fact, I think that the "naive comprehension axiom" is more fundamentally problematic than Russel's "paradox".

The problem is that it tries to define any collection by a property of member.

For example, let us think of the collection of some balls I put into a bag.

I put a ball into the bag just because my groping hand happened to touch the ball, not because the ball is red or blue or something.

Special-Student-7-Rebutter
You mean that the collection is not defined by any property of ball, like being red or something?

Special-Student-7-Hypothesizer
Yes.

Special-Student-7-Rebutter
Your collection is usually defined by enumerating the balls, as the collection is finite: you cannot put infinitely many balls into the bag by groping.

Special-Student-7-Hypothesizer
Yes, when the collection is finite, there is the escape.

But what if the collection is infinite?

Special-Student-7-Rebutter
How will you define an infinite collection not by property of member?

Special-Student-7-Hypothesizer
If we ignore Relativity, let us take a ${\mathbb{R}}^3$ Cartesian coordinates system for the Universe, and take the 1-radius open ball around each rational-coordinates point. Then, the numbers of the electrons in each open ball at a time, which is a subcollection of the natural numbers set, is my infinite collection.

Special-Student-7-Rebutter
You mean that the collection is not defined by any property of natural number, like being even or something?

Special-Student-7-Hypothesizer
Yes.

Special-Student-7-Rebutter
It does not seem to be really infinite, supposing that there are only some finite number of electrons in the Universe.

Special-Student-7-Hypothesizer
Then, let us take the pair of the center-coordinates and the number of the electrons, like ((1.2, 3.4, 5.6), 7).

Special-Student-7-Rebutter
Then, certainly, the collection is infinite.

What if we are not allowed to ignore Relativity?

Special-Student-7-Hypothesizer
Let us take a chart for the spacetime manifold, whose (the chart's) domain is homeomorphic to ${\mathbb{R}}^4$, and let us do the same thing for that ${\mathbb{R}}^4$.

Special-Student-7-Rebutter
So, you are saying that such collections exist in the reality but the "naive comprehension axiom" cannot grasp such collections.

Special-Student-7-Hypothesizer
In fact, that is also the problem of the "restricted comprehension axiom", which is in fact the same with the "naive comprehension axiom" except that the "restricted comprehension axiom" thinks of only the elements of an already-known-to-be-set collection.

Special-Student-7-Rebutter
Is "the number of the electrons in the open ball" not 'property of member'?

Special-Student-7-Hypothesizer
At least, the ZFC set theory does not admit such a property: the "restricted comprehension axiom" requires that the property is expressed as a formula that allows only the specified operators like $\in$ and some already-known-to-be-set collections.

Special-Student-7-Rebutter
Did the "naive comprehension axiom" admit such a property?

Special-Student-7-Hypothesizer
Maybe, but such a usage of the term, "property of member", would be quite harmful (a too-far-fetched interpretation, I would say), in my opinion. Being "the number of the electrons in the open ball" is not any property of the natural number but is a property of the Universe: I mean, being "the number of the electrons in the open ball" is not about the natures of the natural number (like being even, prime, or something) but about the natures of the Universe.

Special-Student-7-Rebutter
Anyway, the "restricted comprehension axiom" does not allow your collection.

Special-Student-7-Hypothesizer
So, there is the collection in the reality, but the ZFC set theory refuses to cope with the collection.

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5: "shugo" Means 'Collection'

Special-Student-7-Hypothesizer
So, "shugo" in Japanese textbooks means 'collection' not "set".

And when a Japanese student later hears of Russel's "paradox" and hears '"set" of some 2 electrons' refuted, the "shugo" of the 2 electrons is not refuted at all, because the collection of the 2 electrons is not refuted at all.

And when he or she later hears that "naive set theory" is inappropriate, the concept of "shugo" (the concept of 'collection') is not inappropriate at all, because the concept of 'collection' is not based on the naive comprehension axiom.

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6: The Concept of 'Collection' Lives on

Special-Student-7-Hypothesizer
The concept of 'collection' is still valid and is indeed used in mathematics.

As a typical example, in the category theory, a category in general is not any set but a collection. For example, the category of all the sets, $Set$, is a collection but not any set.

Special-Student-7-Rebutter
$Set$ is prevalently called "class".

Special-Student-7-Hypothesizer
Yes, 'class' is a concept wider than 'set' and narrower than 'collection', but anyway, a class is not any set in general.

Special-Student-7-Rebutter
How is 'class' narrower than 'collection'?

Special-Student-7-Hypothesizer
Roughly speaking, any class is still constructed from the empty set and is still defined by property of member with a formula.

Anyway, if something was inappropriate just because it was not "set" in the ZFC set theory, the whole category theory would be inappropriate.

And "collection" is still frequently used in many mathematical textbooks: while a reader may wonder why "collection" is being used instead of "set", that is because the mentioned object is not or is not necessarily any "set" according to the ZFC set theory, and if 'collection' was inappropriate, such textbooks would be inappropriate, which is not the case.

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7: Mathematics Is Not Exactly Built on the Set Theory

Special-Student-7-Hypothesizer
A rather prevalent misconception is that "whole mathematics is built on the set theory.".

Special-Student-7-Rebutter
That is what the naive set theory aspired to do.

Special-Student-7-Hypothesizer
But the attempt failed (by mainly Russel's "paradox") and because the set theory gave up dealing with general collections and has degenerated to be a theory on a very limited kind of collections, the aspiration is given up.

Certainly, the set theory is still a very important part of mathematics, but it is not exactly the basis of whole mathematics.

Special-Student-7-Rebutter
A proof is that "collection" is still used in many mathematical textbooks.

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References

<The previous article in this series | The table of contents of this series | The next article in this series>

3: What Are Discrete? What Are Continuous?

<The previous article in this series | The table of contents of this series | The next article in this series>

Water is not continuous, because it is made of molecules. Even if we ignore the fact, "Water is continuous." is mathematically nonsense.

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Topics

About: elementary school mathematics

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Water Is Not Continuous
• 2: Continuousness Cannot Be Defined as a Property of Set
• 3: Continuousness is About Measure
• 4: What Is Discrete Measure?
• 5: Continuousness Is Not About Covering a Line
• 6: Continuousness May Be About Map
• 7: What Should Be Taught to Low-Level Schools Students

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Starting Context

• The reader knows the background of this site.

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Target Context

• The reader will know what 'discrete' or 'continuous' means in mathematics.

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Orientation

There is an article on becoming a benefactor of humanity by being a conduit of truths

---

Main Body

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1: Water Is Not Continuous

Stage Direction
Special-Student-7-Hypothesizer has a paperback in his hands, turning over some pages back and forth.

Special-Student-7-Hypothesizer
This book, an enlightening book on mathematics (a good one in that it arouses interests in mathematics, whatever we say about it hereafter), says that there are discrete amounts and continuous amounts, and cites water as an example of continuous amounts.

Special-Student-7-Rebutter
I do not understand a bit: while any chunk of water is a set of molecules, how is water a "continuous amount"?

Special-Student-7-Hypothesizer
Obviously, the author ignored the fact that water was made of molecules.

Special-Student-7-Rebutter
Then, what water was supposed to be?

Special-Student-7-Hypothesizer
Water seems to be have been supposed to be what the ancients imagined to be.

Special-Student-7-Rebutter
What did the ancients imagine?

Special-Student-7-Hypothesizer
Well, ... I do not know how to state it.

Special-Student-7-Rebutter
But modern mathematics says that the chunk of water is made of some points anyway, even if the points are not suppose to be the molecules, does it not?

Special-Student-7-Hypothesizer
I think so: the only way of describing the chunk of water by modern mathematics is to regard the chunk of water as a set of some points, whether the set is a set of finite number of molecules or a set of infinite number of points, as far as I know.

Special-Student-7-Rebutter
And how is the set of infinite number of points "continuous amount"?

In the 1st place, what does "amount" mean? Is the set itself "amount", is the number of points "amount", is the volume of the chunk "amount", is the mass of the chunk "amount", or what?

Special-Student-7-Hypothesizer
The book seems to be saying that the chunk of water itself is a "continuous amount".

Special-Student-7-Rebutter
I do not understand a bit.

Special-Student-7-Hypothesizer
The book compares the chunk of water with a set of apples and says that water is continuous because while each element of the apples set is separated and independent from the other elements, that is not the case for the chunk of water.

Special-Student-7-Rebutter
I disagree. Each point of water is separated from any other point: for example, denoting any point as $(x,y,z)$, any other point will be $(x^{\prime },y^{\prime },z^{\prime })$ such that $(x,y,z)\ne (x^{\prime },y^{\prime },z^{\prime })$, and while the distance of the 2 points, $\sqrt{(x^{\prime }-x{)}^2+(y^{\prime }-y{)}^2+(z^{\prime }-z{)}^2}$, is positive, how are the 2 points not separated? Besides, what does "independent" mean? Any distinct 2 points of any set should be independent, because they are different points: otherwise, they would be the same point.

Special-Student-7-Hypothesizer
What the author wanted to say seems that between any distinct points of water, there is another point of water.

Special-Student-7-Rebutter
That is the case also for the rational numbers set (I am not talking about the real numbers set) with the canonical order, but mathematics does not call the rational numbers set "continuous", as far as I know.

Special-Student-7-Hypothesizer
The book also says like "Water can be divided infinite times to still be water and when any 2 chunks of water are united, the union is seamless.".

Special-Student-7-Rebutter
I sincerely disagree: any single point of water cannot be divided, while any infinite set of apples can be divided infinite times: for example, supposing that the apples are indexed with the natural numbers, divide the set into the even-indexed set and the rest; divide the even-indexed set into the 4-multiple-indexed set and the rest; and so on; seamless? but the distinction of whether a point is from the 1st chunk or from the 2nd chunk still exists, and I call the distinction "seam": it is just a matter of that we think of the imaginary partition board in the united water.

Special-Student-7-Hypothesizer
Certainly, whether a set can be divided infinite times is just about whether the set is infinite; whether a set is seamless is just about whether the union is of sets with some common points: if there is no common point, the union has the seam defined by the origins of the points.

Special-Student-7-Rebutter
Then, how is the chunk of water "continuous amount"?

Special-Student-7-Hypothesizer
The book also says that the elements of the apples set can be counted, while the elements of the water chunk set cannot be counted.

Special-Student-7-Rebutter
Huh? Are we still talking about continuousness? That was about the distinction between countable sets and uncountable sets.

Special-Student-7-Hypothesizer
Mathematics distinguish countable sets and uncountable sets but does not call uncountable sets "continuous".

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2: Continuousness Cannot Be Defined as a Property of Set

Special-Student-7-Hypothesizer
Continuousness cannot be defined as a property of set, because any pure set is just a collection of elements without any relation between elements assumed.

Special-Student-7-Rebutter
The word, "continuous", insinuates relations between elements, but the concept of 'set' has eliminated relations between elements.

Special-Student-7-Hypothesizer
In mathematics, abstraction is crucial: once any concept has been defined, it is important that any undue property that has been eliminated from the concept is never surreptitiously assumed.

Special-Student-7-Rebutter
The statement, "between any distinct points of water, there is another point of water", is not about any pure set, but about a set with an order, while the order is an extra structure put onto the set.

Special-Student-7-Hypothesizer
To distinguish concepts clearly is the 1st step for clear thinking, and mathematics is arguably the best way for teaching making clear distinctions of concepts, but lower-level schools are missing the opportunity.

In fact, whether one is good with mathematics is not about being good with numbers (mathematics is not always about numbers), but about being good with making clear distinctions of concepts.

In fact, arguing with a mathematics-hater is very futile, because he or she incessantly confuses things and his or her arguments are just muddles.

Special-Student-7-Rebutter
And there are so many mathematics-haters ...

Special-Student-7-Hypothesizer
Continuousness is not about set or about set with order: certainly, we can think of whether another element exists between any 2 distinct elements in a set with an order, but mathematics does not call that property "continuousness".

Special-Student-7-Rebutter
I understand that all cannot be taught in lower-level schools, but I claim that no lie should be taught in whatever level.

Special-Student-7-Hypothesizer
Not teaching something is OK, but teaching a lie is not OK, while "Water is continuous." is a lie.

Special-Student-7-Rebutter
Such lies may be "easy" for many unenthusiastic students, but sincere students will be confused by such lies, because lies do not make sense if sincere students contemplate them.

Special-Student-7-Hypothesizer
Education on the Bias Planet is giving priority to pampering unenthusiastic students over not-confusing sincere students, because unenthusiastic students are many while sincere students are very few: it is called "democracy".

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3: Continuousness is About Measure

Special-Student-7-Hypothesizer
In fact, continuousness is about measure.

Special-Student-7-Rebutter
What is 'measure'?

Special-Student-7-Hypothesizer
Any measure is an extra structure put onto a set that (the extra structure) is defined like this: we think of a $\sigma$-algebra, which is a subset of the power set (the set of all the subsets of the set) of the set, and map to each element of the $\sigma$-algebra an element of $[0,\mathrm{\infty }]$, while the $\sigma$-algebra and the map have to satisfy certain conditions.

That may sound enigmatic to many students, but for example, the method of taking volumes of water chunks is a measure.

The reason for choosing a $\sigma$-algebra instead of using the power set itself is that we do not need to measure all the subsets (so, we call each element of the $\sigma$-algebra "measurable subset").

Special-Student-7-Rebutter
That is the reason why I asked what "amount" meant: "Water is continuous." is nonsense because any measure is not specified.

Special-Student-7-Hypothesizer
Note that we can think of many measures for the same set: we can take a measure that takes volumes, a measure that takes masses, the measure that counts numbers of points (called "counting measure"), and so on; if we choose different units of volumes or masses, the measures are different.

Special-Student-7-Rebutter
Can we take the counting measure for the chunk of water while the set is uncountable?

Special-Student-7-Hypothesizer
Yes we can. The counting measure maps to each infinite measurable subset $\mathrm{\infty }$ and maps to each finite measurable subset the number of the points, so, the set's being uncountable does not prevent the counting measure from being valid.

Special-Student-7-Rebutter
So, although the book is as though the chunk of water cannot use the counting measure, that is not true.

Special-Student-7-Hypothesizer
That is certainly not true, although most people do not use the counting measure for daily purposes.

Special-Student-7-Rebutter
And how is a measure 'continuous'?

Special-Student-7-Hypothesizer
Any measure is continuous if and only if each single point subset measures 0.

Special-Student-7-Rebutter
That is rather unexpectedly simple.

Special-Student-7-Hypothesizer
The volume measure for water is indeed continuous, because the volume of each single point subset is 0.

On the other hand, the counting measure for the chunk of water or the apples set is not continuous, because each single point subset measures 1.

Special-Student-7-Rebutter
So, "Water is continuous." is indeed a nonsense, because the chunk of water can have a non-continuous measure.

Special-Student-7-Hypothesizer
As continuousness is about measure, we need to specify what measure we have chosen.

What the book should have said is "For water, humans usually choose continuous measures, practically speaking.", while the reasons why such a measure is continuous the book cites (like "points are separated"; "infinitely divisible"; "elements are uncountable") are all wrong.

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4: What Is Discrete Measure?

Special-Student-7-Rebutter
What is discrete measure? Is every non-continuous measure discrete?

Special-Student-7-Hypothesizer
No.

'Discrete measure' is defined to be any measure such that there is a countable measurable subset whose complement (which is guaranteed to be a measurable subset by the definition of $\sigma$-algebra) measures 0.

Special-Student-7-Rebutter
In the case of the counting measure for the apples set, ...

Special-Student-7-Hypothesizer
As the whole set is countable, the "countable measurable subset" can be taken to be the whole set, and the complement is the empty set, which measures 0. So, the counting measure for the apples set is discrete.

Special-Student-7-Rebutter
So, the counting measure for each countable set is discrete.

How about the counting measure for the chunk of water?

Special-Student-7-Hypothesizer
There is no such a countable subset, so, is not discrete.

Special-Student-7-Rebutter
So, the counting measure for the chunk of water is not continuous or discrete.

Special-Student-7-Hypothesizer
Also the volume measure for the chunk of water is not discrete.

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5: Continuousness Is Not About Covering a Line

Special-Student-7-Rebutter
What the word, "continuous", reminds most people of may be 'covering a line'.

Special-Student-7-Hypothesizer
Ah, but mathematics does not call it "continuous", as far as I know.

And note that "divisible infinite times" or "having a point between any 2 points" does not imply 'covering a line'.

In fact, the rational numbers set on a line is 'divisible infinite times' and 'has a point between any 2 points', but it does not cover the line.

And note that the real numbers set is just a set and the points do not have inherent locations: if you imagine that the real numbers set is on a line, that is just because you are imagining so: you can also imagine the real numbers scattered around the universe.

Special-Student-7-Rebutter
When you talk about the real numbers set on a line, you are talking about the Euclidean metric space, not about the pure real numbers set.

Special-Student-7-Hypothesizer
Mathematics talks about 'connected metric (or topological) space' but not about "continuous metric (or topological) space", as far as I know.

Any connected topological space (any metric space is canonically a topological space) is defined to be a topological space that is not the union of any disjoint nonempty open subsets, which may not be understood by people who have not learned topology, but you should at least know that criteria like "divisible infinite times" do not define 'connectedness'.

Of course, 'covering a line' is an important issue that historically prompted real numbers, but that is different from being continuous.

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6: Continuousness May Be About Map

Special-Student-7-Rebutter
There is also the concept of 'continuous map', right?

Special-Student-7-Hypothesizer
Yes, we have talked about continuous measure so far because the book should have meant it, but "continuous" is used in some other cases.

"continuous map" is arguably the most famous one of them.

Special-Student-7-Rebutter
What does that mean?

Special-Student-7-Hypothesizer
Continuousness is in fact talked about each point of the domain of the map, and means that for each neighborhood of the value point on the codomain, a neighborhood of the domain point is mapped into the codomain neighborhood (the knowledge of 'topology' is required in order to understand that). Continuous map means that the map is continuous at each domain point.

Special-Student-7-Rebutter
How about "discrete map"?

Special-Student-7-Hypothesizer
I have not heard that concept.

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7: What Should Be Taught to Low-Level Schools Students

Special-Student-7-Rebutter
But what should be taught to low-level schools students? I mean, should they be taught the exact conditions for $\sigma$-algebra or measure?

Special-Student-7-Hypothesizer
I do not particularly say so.

In fact, the concept, 'continuous', does not need to be introduced to them.

Special-Student-7-Rebutter
Does it not?

Special-Student-7-Hypothesizer
The real purpose is just to convince them that natural numbers are not enough; they need fractions and decimals. So, just convince them so, without introducing 'continuousness'.

Special-Student-7-Rebutter
Certainly, 'continuousness' is not indispensable for introducing fractions and decimals.

Special-Student-7-Hypothesizer
As an example, line segment length is better than water volume, because water really consists of molecules, which complicates the situation.

Special-Student-7-Rebutter
But students need to know about water volume anyway.

Special-Student-7-Hypothesizer
Students definitely need to know about approximations. While water really consists of molecules, we make an approximation when we measure the volume of any chunk of water.

Special-Student-7-Rebutter
Yes, 'approximation' and 'error' may be the most important concepts that even low-level schools students need to learn.

Special-Student-7-Hypothesizer
While any chunk of water is made of some molecules and the molecules are moving around, what is indeed the volume of the chunk of water? ... However we exactly define 'volume', the volume is really wobbling, and your measured volume has an error, partly because the volume is wobbling and partly because your act of measuring is inaccurate.

Special-Student-7-Rebutter
It is crucial to understand that that does not mean that the reality itself is ambiguous: you cannot say that the reality is ambiguous just because you cannot measure the reality accurately.

Special-Student-7-Hypothesizer
Unfortunately, that is what is not understood even by many renowned scientists.

Special-Student-7-Rebutter
What should be taught about 'set'?

Special-Student-7-Hypothesizer
Any set is just a collection of some elements with no extra property assumed, and when you measure a set, you need to define an extra structure, which even low-level schools students need to learn.

Special-Student-7-Rebutter
At least, the students need to understand that the chunk of water does not automatically dictate any measure: there can be a volume measure, a mass measure, the counting measure, and a fancy measure someone can concoct, and they need to specify the measure they are going to talk about.

Special-Student-7-Hypothesizer
That is really a matter of of course: without our choosing how to measure the chunk of water, the value of the measuring does not appear, but that knowledge seems to be supposed to be difficult to be understood.

Special-Student-7-Rebutter
As there can be multiple measures for a set, the concept, 'set', has to be established independent of any measure. Without understanding that way of thinking, mathematics cannot be really understood.

Special-Student-7-Hypothesizer
In fact, that way of thinking is the thing that students most need to learn, but they are being pampered on the excuse, "They won't understand anyway.".

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References

<The previous article in this series | The table of contents of this series | The next article in this series>

2: Why Are Mirror Images Regarded to Be Congruent in Plane Geometry?

<The previous article in this series | The table of contents of this series | The next article in this series>

Just a definition by a mathematician? But there should be some good reasons for adopting the definition.

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Topics

About: junior high school mathematics

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Why Are Mirror Images Regarded to Be Congruent in the Plane Geometry?
• 2: It Seems More Natural to Not Call Mirror Images 'Congruent', Supposing That 'Congruent' Means Being Same-Shaped and Same-Sized
• 3: Some Reasons Why We Should Think of Map Instead of "Move"
• 4: 2 Figures Are Congruent iff 1 of Them Is the Image of the Other Under an Isometry
• 5: Isometry Is Really Any Combination of Translation, Rotation, and Mirroring
• 6: But Anyway, Why Is Isometry Used in the Definition of 'Congruence'?
• 7: Mirroring on a Plane or in a 3-Dimensional Space as a Rotation in the Ambient Space

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Starting Context

• The reader knows the background of this site.

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Target Context

• The reader will know some reasons why mirror images are regarded to be congruent in the plane geometry.

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Orientation

There is an article on becoming a benefactor of humanity by being a conduit of truths

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Main Body

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1: Why Are Mirror Images Regarded to Be Congruent in the Plane Geometry?

Special-Student-7-Rebutter
In the (2-dimensional) plane geometry, 2 figures one of which is a mirror image of the other seems to be regarded to be congruent, at least in the Japanese junior high school mathematics.


Special-Student-7-Hypothesizer
The Japanese junior high school mathematics seems to be saying that 2 figures one of which can be "moved" to coincide with the other is regarded to be congruent.

Special-Student-7-Rebutter
What does "move" exactly mean?

Special-Student-7-Hypothesizer
It is the problem that "move" is not defined explicitly in the Japanese junior high school mathematics, although I understand that junior high school students in general are not expected to understand rigorous definitions.

Special-Student-7-Rebutter
But saying "move" is meaningless if the term is not understood accurately.

Special-Student-7-Hypothesizer
Intuitively speaking, a triangle is imagined like a physical object, which is moved in the space: as normal junior high school students cannot teleport objects, they will imagine continuously moving the object in the space, without deforming the object.


Special-Student-7-Rebutter
I see 2 problems there.

1st, how can you be sure that the object is not being inadvertently deformed while you are moving the object?

Special-Student-7-Hypothesizer
Well, the physical object is supposed to be very hard and be never deformed unintentionally by my just moving it.

Special-Student-7-Rebutter
Is "very hard" a legitimate mathematical concept?

Special-Student-7-Hypothesizer
"very hard" is a physical assumption that if the object is wooden or something, the object will not be deformed, at least much, by being moved without being intentionally tortured. ... I know that it is not a refined mathematical concept, but it seems to have been a historical motivation for thinking of 'congruence'.

Special-Student-7-Rebutter
2nd, what does "the space" mean when you say "is moved in the space"? As we are talking about a plane, "the space" should be naturally construed to be the plane, I think. Then, a figure cannot be "moved" to coincide with its mirror image by moving the figure continuously in "the space".

Special-Student-7-Hypothesizer
Apparently, the Japanese junior high school mathematics "moves" the figure in the ambient 3-dimensional space.


Special-Student-7-Rebutter
I do not say that they cannot introduce the ambient 3-dimensional space, but it seems unwise: while we should have been able to concentrate on the 2-dimensional plane, why do they have to complicate the situation by introducing the extra dimension?.

Special-Student-7-Hypothesizer
They seem to have cursorily introduced the ambient 3-dimensional space, because they are familiar with the 3-dimensional space. However, they will have to introduce the ambient 4-dimensional space for the 3-dimensional space geometry, if they are consistent with the strategy.

Special-Student-7-Rebutter
Certainly, 3-dimensional figures can be "moved" in the ambient 4-dimensional space, but it seems to be against the purpose of being intuitive: why do they have to make students think of the hard-to-imagine 4-dimensional space while we are concerned only with the 3-dimensional space?

Special-Student-7-Hypothesizer
Introducing the ambient space seems to be a strategy that is natural only for the plane geometry, and employing a strategy that lacks generality seems unwise.

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2: It Seems More Natural to Not Call Mirror Images 'Congruent', Supposing That 'Congruent' Means Being Same-Shaped and Same-Sized

Special-Student-7-Hypothesizer
In fact, supposing that 'congruent' means being same-shaped-and-same-sized, calling mirror figures 'same-shaped' seems unnatural, in the first place.

Special-Student-7-Rebutter
Are there some grounds for that supposition?

Special-Student-7-Hypothesizer
At least most general (not mathematical) dictionaries say that 'congruent' means being same-shaped-and-same-sized.

Special-Student-7-Rebutter
Mirror images are not same-shaped in my vocabulary.

Special-Student-7-Hypothesizer
If someone feels mirror images being same-shaped, the reason seems to be that he or she has imagined moving figures in the ambient higher-dimensional space.

Special-Student-7-Rebutter
But most people do not naturally imagine moving 3-dimensional figures in the ambient 4-dimensional space, I guess.

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3: Some Reasons Why We Should Think of Map Instead of "Move"

Special-Student-7-Hypothesizer
In fact, introducing "move" seems not wise, while "move" is understood as 'continuously move', as is naturally so.

A reason is that a figure has to be taken out of the plane into the ambient 3-dimensional space, if mirroring wants to be regarded to be a kind of "move".

Special-Student-7-Rebutter
As has been discussed above.

Special-Student-7-Hypothesizer
Another reason is that we have to think of the entire continuous movement of the figure in order for us to talk in terms of "move", while we are interested only in the original figure and the terminal figure.

So, why do we not skip the unnecessary middle?

Special-Student-7-Rebutter
If there is a reason, it seems to be that they are not freed from the intuitive mental picture of moving a physical object.

Special-Student-7-Hypothesizer
But in order for us to guarantee that the figure is not deformed in being moved, we have to guarantee that the figure is not deformed at each position, and then, why do we not guarantee that the figure is not deformed just at the terminal position?

Special-Student-7-Rebutter
I do not guess that there is a legitimate reason.

Special-Student-7-Hypothesizer
So, let us think of 'map' instead of "move", where 'map' means mapping each point of the original figure to a point of the terminal figure, without bothering to move the point continuously in the space.

The merits are the reverse of the unwiseness of introducing "move": we do not need the ambient space and we do not need to think of the middle.

Any map is prevalently denoted like $f:S_1\to S_2$, where $S_1$ and $S_2$ and some sets, and $S_1$ is called 'the domain of the map' and $S_2$ is called 'the codomain of the map'. The map does not need to really map the domain to the whole codomain, and the really mapped part of the codomain is called 'the range of the map'; if the range equals the codomain, the map is called to be a surjection. If any different 2 elements of $S_1$ are mapped to different elements, the map is called to be an injection. Any map that is a surjection and an injection is called a bijection.

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4: 2 Figures Are Congruent iff 1 of Them Is the Image of the Other Under an Isometry

Special-Student-7-Hypothesizer
In fact, mathematically speaking, 2 figures are congruent iff 1 of them is the image of the other under an isometry, where any isometry is any map that preserves the distance between any 2 points.

In the plane geometry, with the sets of 2 figures denoted as $S_1^{\prime }\subseteq {\mathbb{R}}^2$ and $S_2^{\prime }\subseteq {\mathbb{R}}^2$, $S_1^{\prime }$ and $S_2^{\prime }$ are congruent if and only if there is an isometry, $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$, such that $f(S_1^{\prime })=S_2^{\prime }$.

Special-Student-7-Rebutter
Are mirror images congruent by the definition?

Special-Student-7-Hypothesizer
Yes: the lengths of any corresponding sides of any 2 mirror triangles are the same.

Note that "preserves the distance between any 2 points" means that with the distance between 2 points, $p_1,p_2$, denoted as $dist(p_1,p_2)$ and the image of $p_i$ denoted as $f(p_i)$, $dist(f(p_1),f(p_2))=dist(p_1,p_2)$.

Special-Student-7-Rebutter
Is any angle automatically preserved?

Special-Student-7-Hypothesizer
Yes. Although the definition of isometry talks only about lengths, any triangle, $ABC$, is mapped to $A^{\prime }{B}^{\prime }{C}^{\prime }$, and the angle, $\mathrm{\angle }ABC$, equals the angle, $\mathrm{\angle }{A}^{\prime }{B}^{\prime }{C}^{\prime }$, because the side length, $CA$, equals the side length, $C^{\prime }{A}^{\prime }$, in addition that $AB=A^{\prime }{B}^{\prime }$ and $BC=B^{\prime }{C}^{\prime }$.

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5: Isometry Is Really Any Combination of Translation, Rotation, and Mirroring

Special-Student-7-Hypothesizer
Any isometry is really any combination of translation, rotation, and mirroring.




Special-Student-7-Rebutter
Well, is that obvious?

Special-Student-7-Hypothesizer
Although we do not show any rigorous proof, it is intuitively obvious.

Any translation preserves distances.

Any rotation preserves distances.

Any mirroring preserves distances.

And any isometry has to be a combination of translation, rotation, and mirroring, because when a triangle, $ABC$, is mapped to $A^{\prime }{B}^{\prime }{C}^{\prime }$ by the isometry, $A^{\prime }{B}^{\prime }{C}^{\prime }$ is same-shaped-and-same-sized with or a mirror image of $ABC$ (what else can it be indeed?), and $A^{\prime }{B}^{\prime }{C}^{\prime }$ has to have been mapped from $ABC$ by a combination of translation, rotation, and mirroring.

Special-Student-7-Rebutter
We are supposing that if $A^{\prime }{B}^{\prime }{C}^{\prime }$ is same-shaped-and-same-sized, it has to be a combination of translation and rotation, and if $A^{\prime }{B}^{\prime }{C}^{\prime }$ is a mirror image, it has to be a combination of translation, rotation, and mirroring.

Special-Student-7-Hypothesizer
Although we have not proved the supposition rigorously, it seems intuitively obvious.

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6: But Anyway, Why Is Isometry Used in the Definition of 'Congruence'?

Special-Student-7-Rebutter
I understand the definition of 'congruence' with isometry, but why "isometry"?

Special-Student-7-Hypothesizer
That is the point on which we have begun this article. Is a definition, "2 figures are congruent iff 1 of them is the image of the other under a combination of translation and rotation.", not more natural?

Someone may say that it is just that a mathematician (in fact, Euclid, I guess) just adopted the definition with isometry, but I think that that mentality is not good: mathematics is not any discipline in which students have to blindly accept an unreasonable definition just because an authority made the definition; we can and should ask whether there are good reasons to adopt the definition.

Special-Student-7-Rebutter
What are the good reasons for the definition of 'congruence' with isometry?

Special-Student-7-Hypothesizer
Some typical concerns in the plane geometry is that the length of a line segment equals the length of another line segment and an angle equals another angle, and isometry is sufficient for addressing such concerns.

We are typically concerned with whether an angle $\mathrm{\angle }ABC$ equals another angle $\mathrm{\angle }{A}^{\prime }{B}^{\prime }{C}^{\prime }$, and it does not really matter whether the triangle, $A^{\prime }{B}^{\prime }{C}^{\prime }$, is same-shaped-and-same-sized with or a mirror image of the triangle, $ABC$.

Special-Student-7-Rebutter
Although there may be some cases in which it matters, but it is more economical to distinguish between being same-shaped-and-same-sized and being a mirror image only in such rare cases.

Special-Student-7-Hypothesizer
That is the good reason why we have the concept that is defined with isometry.

Special-Student-7-Rebutter
But there is the issue that whether the word, "congruent", is appropriate to represent the concept.

Special-Student-7-Hypothesizer
As is discussed above, supposing that the general meaning of 'congruent' is being same-shaped-and-same-sized, the mathematical "congruent" is against the general meaning or along the general meaning only by introducing the ambient space. Either way, I feel that that use of "congruent" somehow misleading.

Special-Student-7-Rebutter
To summarize, 2 figures are said to be congruent iff 1 of them is the image of the other under an isometry, and the reason why isometry is used there is that isometry is along our typical concerns of whether 2 line segment lengths are equal and whether 2 angles are equal. Whether "congruent" is a desirable word is an issue, but an mirror image can be regarded to be same-shaped-and-same-sized by introducing the ambient space, although whether introducing such an extra space is wise is another issue.

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7: Mirroring on a Plane or in a 3-Dimensional Space as a Rotation in the Ambient Space

Special-Student-7-Hypothesizer
Although we do not really need the ambient space in order to define 'congruence', "One of the 2 figures can be moved in the ambient space to coincide with the other figure." seems to be a view of congruence, and let us see how any mirroring is a rotation in the ambient space.

A mirroring on a plane will be easy to imagine as a rotation in the ambient 3-dimensional space.

Let us suppose that there is a triangle, $ABC$, on the $0<x$ half of the $x-y$ plane, and $ABC$ is mirrored with respect to the $y$ axis, with the mirror image, $A^{\prime }{B}^{\prime }{C}^{\prime }$ on the $x<0$ half of the plane.

As one will be able to imagine, when $ABC$ is rotated $\pi$ around the $y$-axis in the ambient 3-dimensional $x-y-z$ space, $ABC$ coincides with $A^{\prime }{B}^{\prime }{C}^{\prime }$.


More specifically, when any point, $(x,y,z)$, is rotated $\theta$ around the $y$ axis, the image, $x^{\prime },y^{\prime },z^{\prime }$, is $\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}cos\theta & 0& -sin\theta \\ 0& 1& 0\\ sin\theta & 0& cos\theta \end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$, by a matrix notation. If someone is not familiar with matrix notations, what that means is nothing but $x^{\prime }=cos\theta x-sin\theta z,y^{\prime }=y,z^{\prime }=sin\theta x+cos\theta z$.

Special-Student-7-Rebutter
$y$ is not changed as we are rotating the point around the $y$ axis.

Special-Student-7-Hypothesizer
When $z=0$, which means that the point is on the $x-y$ plane, and $\theta =\pi$, $x^{\prime }=-x,y^{\prime }=y,z^{\prime }=0$, which is indeed the mirroring.

Let us see a mirroring in a 3-dimensional space as a rotation in the ambient 4-dimensional space.

Let us suppose that there is a tetrahedron, $ABCD$, in the $0<x$ half of the $x-y-z$ space, and $ABCD$ is mirrored with respect to the $y-z$ plane, with the mirror image, $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$ in the $x<0$ half of the space.


Special-Student-7-Rebutter
While the mirroring on the plane was done with respect to a line, the mirroring in the 3-dimensional space is done with respect to a plane.

Special-Student-7-Hypothesizer
Yes. In fact, a mirror you use to look at your face is a plane, not a line, right?

Special-Student-7-Rebutter
I have not used any line mirror before.

Special-Student-7-Hypothesizer
More generally, a mirroring in a ${\mathbb{R}}^n$ space is done with respect to a $n-1$-dimensional hyperplane. In fact, the line on the plane is a $1$-dimensional hyperplane.

When $ABCD$ is rotated $\pi$ around the $y-z$ plane in the ambient 4-dimensional $x-y-z-w$ space, $ABCD$ coincides with $A^{\prime }{B}^{\prime }{C}^{\prime }{D}^{\prime }$.

Special-Student-7-Rebutter
How can I imagine "rotated $\pi$ around the $y-z$ plane"?

Special-Student-7-Hypothesizer
Let us think in this way: rotations around the $y$-axis in the $x-y-z-w$ space are 3-dimensional rotations in the $x-z-w$ space.

Special-Student-7-Rebutter
Ah, as only the $y$-axis is fixed, it is free in the $x-z-w$ space.

Special-Student-7-Hypothesizer
Let us restrict the 3-dimensional rotations in the $x-z-w$ space to only the rotations around the $z$-axis, which is what we are talking about.

Special-Student-7-Rebutter
So, they are the rotations with the $y$-axis and the $z$-axis fixed.

Special-Student-7-Hypothesizer
And such any rotation can be parameterized by a single angle, $\theta$, which is the angle of the 3-dimensional rotation in the $x-z-w$ space around the $z$-axis.

When any point, $(x,y,z,w)$, is rotated $\theta$ around that way, the image, $x^{\prime },y^{\prime },z^{\prime },w^{\prime }$, is $\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\\ w^{\prime }\end{array}\right)=\left(\begin{array}{cccc}cos\theta & 0& 0& -sin\theta \\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ sin\theta & 0& 0& cos\theta \end{array}\right)\left(\begin{array}{c}x\\ y\\ z\\ w\end{array}\right)$, by a matrix notation. If someone is not familiar with matrix notations, what that means is nothing but $x^{\prime }=cos\theta x-sin\theta w,y^{\prime }=y,z^{\prime }=z,w^{\prime }=sin\theta x+cos\theta w$.

Special-Student-7-Rebutter
It has to be that because it is the 3-dimensional rotation in the $x-z-w$ space around the $z$-axis while the $y$-axis is already fixed.

$y$ and $z$ are not changed of course.

Special-Student-7-Hypothesizer
When $w=0$, which means that the point is in the $x-y-z$ space, and $\theta =\pi$, $x^{\prime }=-x,y^{\prime }=y,z^{\prime }=z,w^{\prime }=0$, which is indeed the mirroring.

Special-Student-7-Rebutter
So, while most people probably cannot easily imagine rotations in the ambient 4-dimensional space, any mirroring in any 3-dimensional space is indeed a rotation in the ambient 4-dimensional space.

Special-Student-7-Hypothesizer
A way to imagine the rotation in the 4-dimensional space is to think of the projections of the rotation into the $x-y-z,x-y-w,x-z-w,y-z-w$ spaces.





For example, the projection of $(x,y,z,w)$ into the $x-y-w$ space is $(x,y,w)$, and $B:(1,0,0,0)$ and $D:(1,0,1,0)$ map to the same $(1,0,0)$; the projection of $(x,y,z,w)$ into the $y-z-w$ space is $(y,z,w)$, and $B:(1,0,0,0),C:(4,0,0,0),B^{\prime }:(-1,0,0,0),C:(-4,0,0,0)$ map to the same $(0,0,0)$.

Special-Student-7-Rebutter
We have seen the projections, so, what?

Special-Student-7-Hypothesizer
Well, an important thing is that when you look at a 3-dimensional space, you are aware that the space is not the whole space but a projection, which means that each point is not really a point but a set of points with 1 parameter.

Special-Student-7-Rebutter
So, when I extend my finger to touch a point in the projection, I have to be aware that I may not really touch the point in the 4-dimensional space, because my finger is touching the $w=0$ one in the set of points while the point is at $w=2$ for example.

Special-Student-7-Hypothesizer
"Should be touching the point but not really touching the point" may seem supernatural but is not: the point is just at another $w$.

Special-Student-7-Rebutter
I see, ..., but so what?

Special-Student-7-Hypothesizer
Well, beyond that, it is a matter of your imagination; in an analogue, you cannot really see any 3-dimensional object, either: you see 2-dimensional projections and your brain imagines the shape of the 3-dimensional object.

Special-Student-7-Rebutter
So, it should be the same with a 4-dimensional object, you mean.

Special-Student-7-Hypothesizer
I am guessing that it is a matter of practices that you begin to imagine the 4-dimensional object.

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References

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